**Abstract:** |
Let be a compact Riemannian homogeneous manifold, an eigenvalue of the Laplace-Beltrami operator acting on ,
and the corresponding eigenspace. A spherical -eigenmap into the unit sphere of a Euclidean vector space
is characterized by having its components in . A -eigenmap is a harmonic map of constant energy density in the sense of Eells-Sampson.
A conformal -eigenmap is called a spherical minimal immersion; it is an isometric minimal immersion of into with respect to
-times the original metric on . For fixed , the set of all -eigenmaps can be parametrized by a compact convex body
in a finite dimensional -module . Similarly, the set of all isometric minimal immersions can be parametrized by a compact convex body , a linear slice
of by a -submodule . A fundamental problem is to determine the highest weights of the irreducible -components
of and which thereby give the dimensions of and and rigidity. Beyond this quest one aims to understand the geometry
of these moduli via Minkowski-type measures of symmetry developed for general convex sets. The aim of this talk is to define a sequence of such measures and calculate them
in specific instances. This, in particular, will give a new understanding of the "roundness" of the space of minimal -orbits in spheres. |
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